In quadrilateral ABCD, the side lengths are AB = 9 cm, BC = 40 cm, CD = 28 cm and AD = 15 cm. Given that angle ABC is a right angle (triangle ABC is right angled at B), what is the difference (in cm) between the perimeter of triangle ABC and the perimeter of triangle ADC?

Introduction / Context:
This question tests properties of a quadrilateral where one of the internal angles is a right angle, and uses the Pythagoras theorem to compare the perimeters of two triangles formed inside the same quadrilateral. It is a typical geometry question seen in aptitude exams and is useful for revising right triangles and perimeter calculations.

Given Data / Assumptions:

• Quadrilateral ABCD has sides AB = 9 cm, BC = 40 cm, CD = 28 cm and AD = 15 cm.
• Angle ABC is a right angle, so triangle ABC is right angled at B.
• We assume the quadrilateral is simple (no self intersection).
• We need the difference between the perimeters of triangle ABC and triangle ADC.

Concept / Approach:
The key idea is that if a triangle is right angled, the side opposite the right angle is the hypotenuse, and we can find it using the Pythagoras theorem: hypotenuse^2 = (leg1)^2 + (leg2)^2. Once we know the common side AC, we can compute the perimeters of triangles ABC and ADC and then take their difference. Perimeter of a triangle is the sum of its three side lengths.

Step-by-Step Solution:
Step 1: In right angled triangle ABC, angle B = 90 degrees, so AC is the hypotenuse opposite angle B. Step 2: Apply Pythagoras theorem: AC^2 = AB^2 + BC^2. Step 3: Substitute AB = 9 cm and BC = 40 cm. So AC^2 = 9^2 + 40^2 = 81 + 1600 = 1681. Step 4: Therefore AC = square root of 1681 = 41 cm. Step 5: Perimeter of triangle ABC = AB + BC + AC = 9 + 40 + 41 = 90 cm. Step 6: Triangle ADC has sides AD = 15 cm, DC = 28 cm and AC = 41 cm. Step 7: Perimeter of triangle ADC = AD + DC + AC = 15 + 28 + 41 = 84 cm. Step 8: Difference between perimeters = 90 − 84 = 6 cm.

Verification / Alternative check:
We can quickly verify the Pythagorean triple 9, 40, 41. Since 9^2 + 40^2 = 81 + 1600 = 1681 and 41^2 = 1681, the triple is exact, so there is no rounding error. The perimeter values 90 cm and 84 cm are direct sums of integer side lengths, so the difference 6 cm is fully consistent.

Why Other Options Are Wrong:
4 cm: This would require triangle ADC to have perimeter 86 cm, which does not match the actual side lengths 15, 28 and 41.
5 cm: This implies one of the perimeters is miscalculated by 1 cm, which is not true given exact integer values.
7 cm: This would give perimeters differing by 7 cm, which is inconsistent with 90 and 84.
8 cm: This is larger than the actual difference and does not arise from any reasonable miscalculation of sides.

Common Pitfalls:
Students sometimes treat AC as a simple sum or average of AB and BC instead of using the Pythagoras theorem. Another common error is to forget that both triangles share the same diagonal AC and to double count or ignore it for one triangle. Careless arithmetic in squaring or adding 9 and 40 can also lead to incorrect hypotenuse and hence incorrect perimeters.

Final Answer:
The difference between the perimeters of triangles ABC and ADC is 6 cm.